Vibrations in a fluid layer between an elastic or rigid sphere and a concentric rigid or elastic shell

Journal of Fluids and Structures (1990) 4, 203-217

VIBRATIONS IN A FLUID LAYER BETWEEN A N ELASTIC OR RIGID SPHERE A N D A CONCENTRIC RIGID OR ELASTIC SHELL S. N. PATITSASAND A. J. PATITSAS Department of Physics and Astronomy, Laurentian University Sudbury, Ontario P3E 2C6, Canada (Received 30 January 1989 and in revised form 5 July 1989) The modes of vibration are examined for a system consisting of an air or water layer between: (a) a rigid sphere and a concentric rigid shell; (b) a steel sphere and a concentric rigid shell; and (c) a rigid sphere and a concentric steel shell. It is shown that low frequency circumferential modes exist in the air layer about the steel sphere which correspond quite well to the low frequencies of acoustic waves emitted when several steel spheres collide in air. The colliding spheres have the same diameter as the sphere inside the air layer. There is appreciable coupling between the modes originating in the steel sphere or steel shell and those originating in the fluid layer, when the fluid is water. 1. I N T R O D U C T I O N THIS STUDYWAS MOTIVATED by the researches of Leach & Rubin (1978) regarding the acoustic emissions from samples of nearly spherical particles, ranging in diameter from 2.3 cm down to 0-03 cm in the case of the Houston booming sand. In all cases, low frequency emissions were recorded having peak frequencies nearly proportional to the inverse of the particle diameter. For the case of steel spheres, the peak frequencies are approximately 18 times lower than the lowest torsional frequency of vibration in the spheres (Sato & Usami 1962; Schreiber et al. 1973). The corresponding peak wavelengths in the surrounding air, at standard conditions, are slightly larger than one-half the sphere circumference. The ratio of the compressional velocity in steel to that in air at standard conditions is 18.5. Similar results were obtained earlier by Koss & Alfredson (1973) and Koss (1974) in their experimental and theoretical investigations of the sound emitted by two identical steel spheres colliding in air. Hidaka et al. (1986) investigated experimentally the sound emitted by particles flowing on an inclined plate. In the study by Koss and Alfredson it is seen that when a sphere, of any composition, is subjected to a unit impulse of acceleration, the radiated pressure wave in the surrounding fluid is a sinusoidal wave having a wavelength equal to one-half the sphere circumference. This leads to the suggestion that when a solid sphere is accelerated in a fluid, there is a tendency for standing waves to be formed around the sphere. These circumferential waves decay very rapidly as their energy is radiated away in all directions. From figure 4 in the study by Koss & Alfredson (1973), it is seen that such waves barely complete one trip around the sphere before their energy is radiated away. In figure 10 in the study by Hidaka et al. (1986), the pressure waveform is considerably longer when the two spheres collide on a plate and the radiation is limited to the upper-half space. When the accelerated sphere is surrounded by several other spheres, such circumferential modes of vibration can be expected to last for longer time intervals. 0889-9746/90/020203 + 15 $03.00

© 1990 Academic Press Limited

204

S. N. PATITSAS A N D A. J. PATITSAS

V~CUUm

Figure 1. Cross-sectionof the shell of inner radius b and thickness t, of the concentricsphere of radius a with the ideal fluid interposed. A rigorous mathematical description of the study of the modes of vibration in the air around a sphere, surrounded by other similar spheres in a nearly closed packed configuration is exceedingly difficult. That is to say, the fitting of the boundary conditions on the surfaces of the surrounding spheres presents a formidable computational problem. However, the problem is relatively easy to treat if the surrounding spheres are replaced by a shell concentric with the inner sphere. A major cross-section of the sphere, the fluid layer and the surrounding shell are shown in Figure 1. Three cases are considered in this study: (a) the layer of an ideal fluid, air or water, between a rigid sphere and a concentric rigid shell; (b) the layer of an ideal fluid between a steel sphere and a concentric rigid shell; and (c) the layer of an ideal fluid between a rigid sphere and a concentric steel shell. The more general case of the layer between an elastic sphere and an elastic shell was not considered in this study due to the increase in the complexity of the computations. It will be seen below that such a generalization would have no appreciable effect in the case of the air layer. The case of the water layer was added since similar experiments involving colliding particles in water could be performed and since the problem of the vibration of a water layer so bounded could be of intrinsic scientific value. The problem of the vibration of a closed thin elastic shell has been treated by Baker (1961) and by Wilkinson (1966). Preliminary computations tend to show that the radial or thickness modes of vibration in the fluid layer do not exist when the sphere is nonconcentric with the spherical shell, but the circumferential modes of vibration exist. If such is the case, then in a spherical cavity filled with a fluid containing several solid particles, the modes of vibration would be of the circumferential type around the particles. The same could be said regarding modes of vibration in a long cylindrical cavity filled with a fluid and

V I B R A T I O N S IN A S P H E R I C A L F L U I D L A Y E R

205

containing several long rods. One may go further along this path and consider modes of vibration in rock bodies composed of a nearly uniform medium and interspersed particles having elastic constants different from those of the surrounding medium. 2. T H E O R Y The equation of motion of a homogeneous isotropic elastic medium under no external force is (Love, 1944) ~211s (X + 2/.,)VA - / * V x (2to) = p 8t 2 , (1) where us is the displacement and A=V.u,,

(2)

2 t o = V X us.

(3)

The subscript s refers to a solid medium. Later, the subscript F will refer to a fluid medium. The medium is characterized by its density, p, and the Lam6 elastic constants )~, /t. The more familiar constants, the Young's modulus E and the Poisson's ratio o, are defined as follows, o= - 2(Z + / , )

(4)

and E ~-2(1+~)"

(5)

Following the approach by Stratton (1941) in solving the equivalent problem in electromagnetic theory, the following may be written as = - V ~

+ V x A,

(6)

where W and A satisfy the equations, 1 ~21~J

V2~ =

(7)

Cl2 Ot 2 '

1 O2A V2A = - -

(8)

C22 Ot 2 '

V - A = 0.

(9)

The term cx= [(A+21z)/p] x/2 is the compressional velocity, i.e. the velocity of longitudinal waves in the medium, and c2 = (l~/p) v2 is the shear velocity, i.e. the velocity of transverse waves in the medium. Next, one introduces the vector wavefunctions L = V%

M = V × r%

1

N = 7- V × M, K2

(10)

where r is the vector distance from the origin (Figure 1), kz = to~c2 and ~p is the solution to equation (7) with cl replaced by c2. The vector A is then expressed as a linear superposition of M and N, L being absent since A has zero divergence. Then

206

s . N . PATITSAS AND A. J. PATITSAS

from equation (6) the components of u~, omitting q~ dependence, are as follows,

d 1)J,(k2r ) Usrt= --al-~rrJl(klr) + btl(l + r -et-~ryt(klr ) + ~l(l + 1)

(cos 0),

(11)

uso,-- (-a,J'(k'r) + b,( dj,(k2r) + j,(~r)) -etYt(klr)r

+ fl(-drrYt(k2r)d

+ Y'(k2r))) ff-~Pt(cos O),

(12)

us~,l = -(dlk2jl(k2r) + &k2yt(k2r)) d P~(cos 0),

(13)

us I : Usrl~ ~1_Usol ~ 31_usdpl~ '

(14)

us = Z us,,

(15)

l=0

where ]l(klr) and yt(klr) are the spherical Bessel functions of the first and second kind, respectively, P/(cos 0) are the ordinary Legendre polynomials, at and el are arbitrary complex coefficients corresponding to compressional motion, bt, dr, 3~ and & are arbitrary complex coefficients corresponding to shear motion, and kl = w/cl. The factor e -i'°t has been omitted since its presence does not influence the equations which follow. The polar and azimuthal angles 0, tp, are shown in Figure 1. The displacement UF in an ideal compressible homogeneous fluid is, a2UF --1 VP,

3t2

(16)

PF

where PF is the density of the fluid and p is the pressure. The pressure satisfies the scalar wave equation, 1 a2P V2p c2 3t 2 , (17) where cF is the bulk velocity in the fluid. The solution for p without ~ dependence is

Pt = (Ajt(kr) + Btyl(kr))Pt(cos 0), (18) where At and Bt are arbitrary complex constants and k = w/ce. From equation (16), BF, = i,iFrlr all-UFOIO "~- UFdpl~ = ~

UFrl= ~ UFot= ~

k

1

1

Vp,,

(Atj;(kr) + Bly;(kr))Pt (cos 0), d

(A,jt(kr) + Btyt(kr)) -~ Pt (cos 0), UFdpt = O,

(19) (20) (21) (22)

where the prime on the Bessel functions denotes differentiation with respect to the argument, in this case kr

VIBRATIONS IN A SPHERICAL FLUID LAYER

207

The above equations for us, uF, including ~ dependence, have been derived but are not included here since no new eigenfrequencies are obtained by including ~b dependence. This fact was also recognized earlier by Silbiger (1962). The stress components in the solid are given in terms of the components of us as follows: 0 arrr = ZA + 2/* ~r usr, (23) u s o l+O r\ - ~ u,~), Vro = Ia (~--TrU,o - --7-

(10

zr~,=l~ rs~nOOqb usr+-~rUs , -

(24) .

(25)

Upon substituting for Usr , UsO from equations (11)-(15), the expressions for the stress components become "~rrl

:

( a l R l t ( r ) + b t R z l ( r ) + e l S u ( r ) + flS2t(r))et ( c o s 0 ) ,

(26)

~rOt = (aiR31(r) + blR4l(r) + etS31(r) + j~S4,(r)) ~0 Pl (cos 0),

(27)

T,.•t = - ( d t R s , ( r ) + glSs,(r)) ff--~ PI (cos 0),

(28)

Rl,(r) = p/z ((k2r2 _ 2l(l - l ) ) j t ( k l r ) - 4klrj,+l(kxr)),

(29)

R21(r) = ~ ((1 -- 1)jl(k2r) -- k2rjt+l(k2r))2l(l + 1),

(30)

R3t(r) = - ~ 2((l - 1)jl(klr ) -- klr]l+l(klr)) ,

(31)

R4t(r) = - ~ ((k~r 2 - 2(l 2 - 1))jl(k2r) - 2k2rj,+l(k2r)),

(32)

Rst(r) = - ~ ((1 - 1)jt(kzr) - k2rj,+a(k2r))k2r,

(33)

Sit(r) = ~22(( k2r2 - 21(l - 1))yt(kl r) - 4klry,+l(klr)),

(34)

S2,(r) = ~ ((l - 1)yt(k2r) - k2ryt+l(k2r))2l(l + 1),

(35)

S3,(r) = - ~ 2((l - 1)yt(klr) - klryl+a(klr)),

(36)

S4l(r) = - - ~ ((kZgr 2 - 2(I 2 - 1))yt(kzr) - 2 k 2 r y t + l ( k 2 r ) ) ,

(37)

S5/(r) = - ~ 2 ((l - 1)yt(kzr) - k 2 r Y l + l ( k 2 r ) ) k 2 r.

(38)

where

208

s.N.

PATITSAS A N D A. J. PATITSAS

3. A P P L I C A T I O N S Case a: the layer of an ideal fluid between a rigid sphere of radius a and a concentric rigid shell of inner radius b. As is customarily done, a rigid medium is understood to be infinitely rigid. In this simple case, the boundary conditions are

at r = a at r = b

UFr = 0 UFr =

0

(39)

From equation (20) the following equations are obtained j ~ ( k a ) A 1 + y ~ ( k a ) B 1 -- O, j ; ( k b ) A 1 + y ~ ( k b ) B 1 = 0.

(40)

The eigenfrequencies for a specific value of l are obtained by setting the determinant of the coefficients equal to zero. Case b: the layer of an ideal fluid between an elastic sphere of radius a and a concentric rigid shell of inner radius b. In this case, the boundary conditions are,

uF~ = 0 UFr = Us~, r~o = 0,

at r = b, P = ~rr

7rr~ = 0

(41a)

at r = a

(41b,c)

at r = a.

(41d,e)

From equations (20), (11), (18) and (26)-(28) the following equations are obtained for l_>l, j ; ( k b ) A l + y ~ ( k b ) n t = O, -klaj;(kla)a

ka , ka I + l(l + 1)]l(k2a)b t - ~--7-~-_jt(ka)A, - ~

, y l ( k a ) n l = O,

(42) Ru(a)a, + R2t(a)bt -jt(ka)a,

- y l ( k a ) B l = O,

R31(a)a I + R41(a)b I = O.

For l = 0 the fourth equation and the coefficient bo are absent, as can be seen in equations (27) and (30). The coefficients el, 3~, gt in equations (11)-(13) are absent, since they multiply the functions y t ( k r ) which are not allowed inside the sphere. It is noteworthy that the equation resulting from equation (41e) is independent of the other four equations which resulted in equations (42). The characteristic equation derived from equation (41e) gives the eigenfrequencies for the torsional vibrations of the elastic sphere in vacuum (Sato & Usami, 1962), (Schreiber et al., 1973). It has the form (l - 1 ) j l ( k z a ) -- k z a j t + l ( k 2 a ) = 0,

! --> 1

(43)

and is independent of the fluid, as expected. Case c: the layer of an ideal fluid between a rigid sphere of radius a and a concentric elastic shell of inner radius b and outer radius d. The medium for r > d is vacuum. In this case, the boundary conditions are, UFr = 0

UFr = USr, 7frr = O,

P = rrr, "~ro ~- O,

at r = a, ~rO = 0, 'rr~p ~" O

(44a) Vr* = 0 at r = d .

at r = b

(44b) (44c)

209

VIBRATIONS IN A SPHERICAL FLUID LAYER

The equations resulting from the boundary conditions ~:r,--0 at r = b and at r = d separate out from the rest of the equations and give the torsional eigenfrequencies in the elastic shell. From the rest of the boundary conditions the following equations are obtained for l -> 1, j~(ka)A, + y~(ka)B1 = O, (45a)

- k l b j ; ( k l b )at + l(l + 1)jt(k2b )bl - klby~(klb )e, + l(l + 1)yl(k2b )ft kb kb to2pFlt(kb)A, - m--T-p-p-FYl(kb)B1 = 0, • t

t

(45b)

gu(b)at + gEt(b)bt + Su(b)el + SEl(b)fll -- jt(kb)A, - yt(kb)B1 = O,

(45c)

R31(b )at + R4t(b )b, + Sat(b )e, + S41(b )ft = 0,

(45d)

gll(d)at + g2t(d)bt + Su(d)el + S21(d)ft = 0,

(45e)

gal(d)al + g4,(d)bt + S3t(d)el + S4t(d)j~ = 0.

(45f)

For l = 0 equations (45d) and (450 and the coefficients bo, f0 are absent as can be seen from equations (27) and (30). 4. N U M E R I C A L RESULTS 4.1. CASE a The results from setting the determinant of equations (40) equal to zero are shown in Figures 2 and 3. Following the terminology of Bird et al. (1960), we observe the curves A (l = 1), B (l = 2) which correspond to ring (circumferential) modes, and the curves C to H (l = 0 , 1, 2) which correspond to thickness modes. The low frequency circumferential curves A, B slope slowly downwards and decrease as 1/b as b gets very large. The values of ka for the curves A, B near b / a - - 1 are 1.41 and 2.44, respectively. For a sphere of radius a -- 1.142 cm and fluid velocity equal to that in air at standard conditions, the corresponding frequencies are 6.49 kHz and 11-20kHz, respectively. The lowest torsional frequency for a steel sphere of the same radius in vacuum is 125 kHz, being 19 times 6-49 kHz. It is seen that the eigenfrequency for the lowest circumferential mode agrees quite well with the low peak frequencies 20

16

12

8

4 =

B

t.=2

A

0 0

'

1!2

'

1!4

,

'

116

L=I

1~8

m

2.0

b/a

Figure 2. ka versus b/a for l = 0, 1, 2 for modes of vibration of a layer of an ideal fluid between a rigid sphere of radius a and a concentric rigid shell of inner radius b.

210

s. N. P A T I T S A S A N D A. J. P A T I T S A S 12 For curve D

- 0-4 -0"8 -1'2

1-0

,

' 1.1

,

Uric

' 1.2

' 1.3

1"0

,~

0'6

0)

'

' 1-4

'

1.5

E @

u

0"2

tU

c~

.......

O0

c3 - 0 . 2

, ......

._2

ue

-0"6 1"0

'

1[ 1

i

I

112

i

i

1"3

i

1"4

1.2

1-5

For curve

A

ue

0.8

0'4 Mr

0"0 . . . . . . . .

-°%o

i

I

lt2

'

113

'

1'.4

1"5

rta

Figure 3. D i s p l a c e m e n t s ur, u o versus r / a for the g e o m e t r y assumed in Figure 2 with ka given by the intersection o f a vertical line at b / a = 1-5 with the curves A (1 = 1), C (1 = 1), D (l = 1) in Figure 2.

observed by Leach & Rubin (1978) for the case of two steel spheres of radius a = 1.142cm colliding in air. However, the behaviour of plots A, B could be quite different in the limit b / a - - > 1.0 if viscous effects were taken into consideration. We observe that there are no circumferential modes for 1 = 0 as can be seen in equation (21). For l = 0, the expressions for J0 and Y0 are sufficiently simple so that the roots of the characteristic equation are given as follows: nzr k a - ( b / a - 1)"

(46)

We also observe that the various thickness curves corresponding to different values of l joint together for large values of ka. This can be shown by using the asymptotic expansions for Jt and Yt- The curves in Figure 3 validate the terminology "circumferential modes" and "thickness modes" used above. Clearly the displacement ur is much smaller than Uo for the curve A in Figure 2, while the opposite is the case for the curves C and D.

211

VIBRATIONS IN A SPHERICAL FLUID LAYER 20--

,.

12

.

~

XX'

8~

4

0 1.0

'

' 1.2

'

' 1.4

'

' 1.6

'

1 .'8

2.0

b/a

Figure 4. ka versus b/a for l = 0 for modes of vibration of a layer of water between a steel sphere of radius a and a concentric rigid shell of inner radius b. The interrupted lines correspond to the curves in Figure 2.

4.2. CASE b

Unlike in Case a, in Case b the velocity c~, the Poisson's ratio a, the Young's modulus E in the elastic sphere, as well as the velocity CF and the density Pe in the fluid, air or water, must be specified. With the fluid assumed to be air at standard temperature and pressure, the plots of ka versus b/a, not shown here, are practically the same as in Figure 2. With the fluid assumed to be water at 20°C, the plots of ka versus b/a are shown in Figures 4 to 7 for l in the range 0 to 2. The respective plots from Figure 2 are also superimposed and are shown as interrupted (dashed) lines. In Figure 4, the nearly horizontal portions of the plots at ka ~-10.5, correspond to the lowest spheroidal eigenfrequency in the sphere. For the case of the steel sphere in vacuum, ka equals

20

16

12

1-0

1.2

1-4

I!6 b/a

Figure 5. Same as in Figure 4 with l = 1.

1'.8

2.0

212

s. N. PATITSAS A N D A. J. PATITSAS

1.0 ka = 7.9448,

Ur

0.6 02 0.0 -0.2

uo ~

Ur

-0"6 -1.(

~

0

~

,

0'2

a

0"6

0"4

0"8

10

12

14

1.6

1"0

1.2

1.4

1.6

0.4

..~ E

8

0.2 11r

o.o u0

~" - 0 " 2 a

ka

= 5-4620

-0.4 -0"6

t

0

I

I

0.2

I

h

0'4

i

I

0"6

i

L

0"8

0'2 0"0

..............

f--I

-0"2

Ur

............... , UO

-0.4

ka=1-0099

-0'6 -0"8 -1.0

0'.2

i

0'-4

i

0'.6

i

i

o-a

i

~:0

0

(.2 ' 1!4

i

~.6

rla

Figure 6. Displacements ur, u o versus r/a for the geometry assumed in Figure 4 with ka given by the intersection of a vertical line at b / a = 1.6 and the three lowest curves in Figure 5.

10.820 for the same eigenfrequency. Frequency cutoffs are observed in the neighborhoods of b/a equal to 1.3, 1.6, 1.9. This must be due to the interaction between the vibrations in the sphere and in the water. Similar cutoffs were observed by Bird et al. (1960). For the case of air, the effect of such an interaction was found (not shown here) to be practically absent. Similar results are observed in Figure 5 for l = 1. Three spheroidal eigenfrequencies appear now at ka, equal to about 7.9, 15.8, 18.0. For the case of the steel sphere in vacuum ka has the corresponding values 7.943, 15-896, 17.992. For the lowest spheroidal eigenfrequency, the curve dips to 6-03 at b/a = 1.0, while for the second and third, the curves dip to 9.61 and 16-99, respectively, at b/a = 1.0. These values of ka were found to be the same when the sphere was assumed to be constrained by a rigid shell of radius a. A s in Figure 2, the behaviour of these plots in the limit b/a--~ 1.0 could be different if viscous effects were taken into account. The low frequency circumferential curve dips to zero at b/a = 1-075. In Figure 6 the displacements ur, ue are plotted versus r/a for ka equal to 1-0099, 5-4620 and 7.9448, which are obtained from the intersection of a vertical line at b/a = 1.6 and the three

213

VIBRATIONS IN A SPHERICAL FLUID LAYER 20

16

12

010

'

112

'

1 '4

'

1-'6

t"8

'

2-0

b/a

Figure 7. Same as in Figure 4 with l = 2.

lowest curves in Figure 5. These plots reveal the following: for ka = 1.0099, the circumferential mode originates in the water; for ka = 5-4620, the thickness mode originates in the water; for ka = 7-9448, the mode originates in the sphere. The case for l = 2 is shown in Figure 7. The three spheroidal eigenfrequencies now occur at ka equal to about 6.0, 11.0 and 19.0. For the case of the steel sphere in vacuum, ka has the corresponding values 6-019, 11-273 and 19.251. The low frequency curve, unlike the case with l = 1, rises as b / a , decreases and joins up with the curve representing the lowest spheroidal eigenfrequency in the sphere. We observe a relatively large cut off interval at b/a ~-1-6 for ka ~-6-0 and relatively small cut off intervals at b/a equal to 1.3, 1-5 and 1.8 for ka ~- 11.0. 4.3.

CASE C

The results for this case, with the fluid being air at standard conditions, are shown in Figure 8. All curves slope downwards as b increases, suggesting that the frequencies in the range considered here correspond to circumferential wave propagation in the shell. Both compressional and shear components enter in these waves. The relative amplitudes ur, Uo at b/a = 1-5, ka -- 15-5, for curve D were computed, revealing that both amplitudes enter with approximately equal magnitude on average over the range b<_r<_b+t. The results with the fluid being water at 20°C are shown in Figures 9 to 14. In Figure 9, l is equal to zero and the shell thickness t is equal to 0-1 a. It is observed that the effect of the elasticity in the shell is to slightly lower the eigenfrequencies corresponding to the thickness modes in the water as compared to those in Figure 2, shown by interrupted (dashed) lines here. The lower curve extending from b/a---1.05 to b/a ~- 1.55 represents circumferential vibrations in the shell. It corresponds to curve A in Figure 8 where k = 09/Cv and CF is the velocity of sound in air. In Figure 10, the thickness t is 10-times larger than that in Figure 9. The shell appears more rigid to the water and the curves for the thickness modes in the water coincide with those in Figure 2, except at the cutoff intervals. The nearly horizontal segments at ka ~-13.5 must represent the lowest thickness m o d e in the shell. This follows from a rough estimate of the wavelengths for the circumferential and thickness waves in the shell.

214

s.N.

P A T I T S A S A N D A . J. P A T I T S A S

20

16

12

B 8

4

°1.o

'

L

1:2

i

114

116

'

118

2.0

b/a

Figure 8. ka versus b/a for the modes of vibration of a layer of air between a rigid sphere of radius a and a concentric steel shell of inner radius b a n d thickness t. The curves corresponding to the vibrations in air are the same as in Figure 2 and are omitted. The shell thickness t and the value of I for each of the curves A to G are as follows: A(l=O,t=O.la), B(l=O,t=l.Oa), C(l=l,t=O.la), D(l=l,t=l.Oa), E ( l = 2 , t = 0.1a), F ( l = 2 , t = 1.0a), G (same as F).

The curves in Figure 11, for l = 1, are very similar to those in Figure 9 for l = 0, the shell thickness t being equal to 0-1 a in both cases. It is noteworthy that the low frequency circumferential m o d e in water is absent. In Figure 12, w h e r e the shell thickness is 10-times larger than in Figure 11, this m o d e is present. Apparently, the thin shell lacks the necessary stiffness to support this m o d e . The relative amplitudes ur, uo were c o m p u t e d in the range 0 < r < b + t at b/a = 1.6 for the six lowest curves. Only for the sixth curve with ka = 12-848 is there a node in Ur inside the shell, suggesting that this curve represents a thickness m o d e in the shell. The results in Figure 13 for 1 = 2 are very similar to those in Figure 11. T h o s e in Figure 14 are quite similar to those in Figure 12, except that the low frequency circumferential m o d e in the water is absent for b/a in the range 1-0 to 1-68. 20i

16

12

8

O,

S ~

1!2

i

tj4

t b/a

1.t6

i

118

i

2-0

Figure 9. Same geometry as in Figure 8 with water in place of air, l = 0, shell thickness t = 0-1a. The interrupted (dashed) lines correspond to the curves in Figure 2 for l = 0.

20

16

12

1J2

1~4

'

1.'6

1~8

2-0

bla

Figure 10. Same as in Figure 9 with shell thickness t = 1.0a.

20

16

12

/

O. 0

t

112

'

1:'4

.......

I

....

L

1.6

!

1 8,

2.0

b/a

Figure 11. Same as in Figure 9 with l = 1; shell thickness t = 0-1a.

20

16

12

f ~o

1!2

~

1~-4-

~

1"6

.h

118

L

bla

Figure 12. Same as in Figure 9 with 1 = 1; shell thickness t --- 1-0a.

2"0

216

S. N. P A T I T S A S A N D A. J. P A T I T S A S

20

16

12

'

1.0

lt2

'

114

' b/a

i

1~6

i

118

2"Q

Figure 13. Same as in Figure 9 with 1 = 2 ; shell thickness t = 0-1a.

20

16

1.0

'

1'.2

'

(.4

'

~e

'

1!8

'

2-0

b/a

Figure 14. Same as in Figure 9 with l = 2; shell thickness t = 1-0a.

The case of the steel case of the air layer corresponding to the characteristic equation, the finite steel shell.

shell having infinite outer radius was also considered. For the it was found that the quasi-frequencies in the air layer, minima of the absolute value of the determinant of the are practically the same as the eigenfrequencies in the case of

5. C O N C L U S I O N S For the case of the air layer between a steel sphere and a concentric rigid shell or between a rigid sphere and a concentric steel shell, the fundamental eigenfrequencies of the circumferential modes in the air layer have values very close to the low frequencies emitted when the same steel spheres collide in air. There is practically no interaction between the modes of vibration in the air layer and those in the steel sphere

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or the steel shell. For the case of the air layer between a rigid sphere and a steel shell of infinite outer radius, the minima of the absolute value of the determinant of the characteristic equation occur practically at the same values of the w a v e n u m b e r , as do the zeros of the same d e t e r m i n a n t when the infinite medium is rigid. This leads to the suggestion that the frequencies of the quasi-modes of vibration, when the air layer is bounded externally by an array of steel spheres, are close to the eigenfrequencies in the air layer bounded by a rigid or steel shell. For the case of the water layer there are also low frequency circumferential modes of vibration. There is appreciable interaction between the vibrations in the steel sphere or steel shell and the vibrations in the water layer. This results in the decrease of the eigenfrequencies in the water layer and the removal of several of the circumferential modes in some cases. T h e r e are frequency cut offs in the regions where the eigenfrequencies for the modes originating in the steel sphere or steel shell coincide with those in the water layer. A t the cut off intervals, the determinant of the characteristic equation does not vanish as it would if there was no interaction between the vibrations in the steel and in the water media. The cut off intervals are more pronounced when the radial displacement in the steel m e d i u m tends to have a maximum value at the water boundary. REFERENCES

BAKER, W. E. 1961 Axisymmetric modes of vibration of a thin spherical shell. Journal of the Acoustical Society of America 33, 1749-1758. BIRD, J. F., HART, R. W. & McCLuRE, F. T. 1960 Vibrations of thick-walled hollow cylinders: exact numerical solutions. J. Acoustical Society of America 32, 1404-1412. HIDAKA, J., INOUE, K. & MIWA, S. 1986 Frequency spectrum of the sound radiated from the flow of granular materials. In Proceedings 1st World Congress on Particle Technology (Ed. K. Leschonski), pp. 289-300. NiJrnberg: NMA Niirnberger Messe-und Ausstellungsegesellschaft mbH Messezentrum. Koss, L. L. & ALFREDSON, R. J. 1973 Transient sound radiated by spheres undergoing an elastic collision. Journal of Sound and Vibration 27, 59-75. Koss, L. L. 1974 Transient sound from colliding spheres, normalized results. Journal of Sound and Vibration 36, 541-553. LEACH, M. F. & RUBIN, G. A. 1978 Size analysis of particles of irregular shape trom their acoustic emissions. Powder Technology 21, 263-267. LOVE, A. E. 1944 A Treatise on the Mathematical Theory of Elasticity. New York: Dover Publications. SATO, Y. & USAMI, T. 1962 Basic study on the oscillation of a homogeneous elastic sphere. Geophysical Magazine (Japan Meteorological Agency, Tokyo) 31, 15-24. SCHREIBER, E., ANDERSON, O. & Sot3A, N. 1973 Elastic Constants and their Measurement. New York: McGraw-Hill. SILBIGER, A. 1962 Nonaxisymmetric modes of vibration of thin spherical shells. Journal of the Acoustical Society of America 34, 862. STRATrON, J. A. 1941 Electromagnetic Theory. New York: McGraw-Hill Book Company. WILKINSON, J. P. D. 1966 Natural frequencies of closed spherical sandwich shells. J. Acoustical Society of America 40, 801-806.